Michael
Harris
Université Paris 7
ARITHMETIC
AUTOMORPHIC FORMS ON UNITARY GROUPS AND RELATED GALOIS REPRESENTATIONS
The
goal of this series of lectures is to present some of the recent
developments in Langlands' program, outlined
more than thirty years ago, to
study the Galois representations attached to certain classes of
automorphic representations
using Shimura varieties. I
will concentrate on the automorphic representations of unitary groups,
since these are most
closely related to GL(n) and lend
themselves to a number of applications. Specific attention will
be devoted to the
ideas and techniques involved in
my recent proof with Clozel, Shepherd-Barron, and Taylor of the
Sato-Tate conjecture
for those elliptic curves over Q that satisfy a certain degeneracy
hypothesis. Recent results on the stabilization of the
Selberg trace formula have
undoubtedly made this hypothesis superfluous, and I will sketch a proof
in the general
case, making use of work in
progress of a number of people.
Rather than attempt to present
complete proofs, I will stress the relations between various approaches
to automorphic
forms on unitary
groups. Whenever possible, I will indicate unsolved
problems that may or may not be accessible
using available techniques.
TENTATIVE
PROGRAM as of 3/7/2008
January 25: The Sato-Tate conjecture for one or two
elliptic
curves, and the relation to L-functions
February 1: Automorphic representations of GL(n) and
their
associated Galois representations: the state of the art
February 8: Automorphic induction and its
opposite:
applications to symmetric powers
February 15: Odd symmetric powers, concluded; the problem of
irreducibility of automorphic Galois representations
February 22: The method of potential automorphy and Calabi-Yau
hypersurfaces
February 29: Deformations of polarized regular Galois
representations
March 7: Hecke algebras for
unitary groups
March 14: The
(Kisin)-Taylor-Wiles method for unitary automorphic forms
April 4: Constructing
automorphic Galois
representations, step 3
(p-adic continuity)
NOTE: THIS LECTURE STARTS AT 1:30 PM
April 11: Constructing
automorphic Galois representations, step 1
(analysis of the stable spectrum of unitary groups)
April 18: Constructing
automorphic Galois representations, step 2
(analysis of the endoscopic spectrum of unitary groups)
April 25: The theta
correspondence as an alternative to step 2, and special values of
L-functions
May 2:
Prospects: topics to be announced.